Location, correlation, radiation: where is the $\sigma$, what is its structure and what is its coupling to photons?

Abstract

Scalar mesons are a key expression of the infrared regime of QCD. The lightest of these is the $\sigma$. Now that its pole in the complex energy plane has been precisely located, we can ask whether this state is transiently $\bar{q}q$ or $\overline{qq}qq$ or a multi-meson molecule or largely glue? The two photon decay of the $\sigma$ can, in principle, discriminate between these possibilities. We review here how the $\gamma \gamma \to {\pi }^{+}{\pi }^{-}$, ${\pi }^0{\pi }^0$ cross-sections can be accurately computed. The result not only agrees with experiment, but definitively fixes the radiative coupling of the $\sigma$. This equates to a two photon width of $(4.1\pm 0.3)$ keV, which accords with the simple non-relativistic quark model expectation for a $\bar{u}u,\bar{d}d$ scalar. Nevertheless, robust predictions from relativistic strong coupling QCD are required for each of the possible compositions before we can be sure which one really delivers the determined $\gamma \gamma$ coupling.

Comments: 18 pages, 11 figures. To be published in Modern Physics Letters A A number of references updated and three sentences changed in the text to reflect these